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Mirrors > Home > MPE Home > Th. List > Mathboxes > submateqlem2 | Structured version Visualization version GIF version |
Description: Lemma for submateq 31074. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
Ref | Expression |
---|---|
submateqlem2.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
submateqlem2.k | ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) |
submateqlem2.m | ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) |
submateqlem2.1 | ⊢ (𝜑 → 𝑀 < 𝐾) |
Ref | Expression |
---|---|
submateqlem2 | ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fz1ssnn 12939 | . . . . . 6 ⊢ (1...(𝑁 − 1)) ⊆ ℕ | |
2 | submateqlem2.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) | |
3 | 1, 2 | sseldi 3965 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
4 | 3 | nnge1d 11686 | . . . 4 ⊢ (𝜑 → 1 ≤ 𝑀) |
5 | submateqlem2.1 | . . . 4 ⊢ (𝜑 → 𝑀 < 𝐾) | |
6 | 4, 5 | jca 514 | . . 3 ⊢ (𝜑 → (1 ≤ 𝑀 ∧ 𝑀 < 𝐾)) |
7 | 3 | nnzd 12087 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
8 | 1zzd 12014 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
9 | fz1ssnn 12939 | . . . . . 6 ⊢ (1...𝑁) ⊆ ℕ | |
10 | submateqlem2.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
11 | 9, 10 | sseldi 3965 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
12 | 11 | nnzd 12087 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
13 | elfzo 13041 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) | |
14 | 7, 8, 12, 13 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) |
15 | 6, 14 | mpbird 259 | . 2 ⊢ (𝜑 → 𝑀 ∈ (1..^𝐾)) |
16 | 2 | orcd 869 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁)) |
17 | submateqlem2.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
18 | nnuz 12282 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
19 | 17, 18 | eleqtrdi 2923 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1)) |
20 | fzm1 12988 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘1) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) |
22 | 16, 21 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
23 | 3 | nnred 11653 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
24 | 23, 5 | ltned 10776 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝐾) |
25 | nelsn 4605 | . . . 4 ⊢ (𝑀 ≠ 𝐾 → ¬ 𝑀 ∈ {𝐾}) | |
26 | 24, 25 | syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝑀 ∈ {𝐾}) |
27 | 22, 26 | eldifd 3947 | . 2 ⊢ (𝜑 → 𝑀 ∈ ((1...𝑁) ∖ {𝐾})) |
28 | 15, 27 | jca 514 | 1 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 {csn 4567 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 1c1 10538 < clt 10675 ≤ cle 10676 − cmin 10870 ℕcn 11638 ℤcz 11982 ℤ≥cuz 12244 ...cfz 12893 ..^cfzo 13034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 |
This theorem is referenced by: submateq 31074 |
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